Eigen 库 和 Matlab 间的简单对应关系
% // A simple quickref for Eigen. Add anything that's missing.
% // Main author: Keir Mierle
% // http://eigen.tuxfamily.org/dox/AsciiQuickReference.txt
%
% #include <Eigen/Dense> or #include <Eigen/core>
%
% Matrix<double, 3, 3> A; // Fixed rows and cols. Same as Matrix3d.
% Matrix<double, 3, Dynamic> B; // Fixed rows, dynamic cols.
% Matrix<double, Dynamic, Dynamic> C; // Full dynamic. Same as MatrixXd.
% Matrix<double, 3, 3, RowMajor> E; // Row major; default is column-major.
% Matrix3f P, Q, R; // 3x3 float matrix.
% Vector3f x, y, z; // 3x1 float matrix.
% RowVector3f a, b, c; // 1x3 float matrix.
% VectorXd v; // Dynamic column vector of doubles
% double s;
%
% // Basic usage
% // Eigen // Matlab // comments
% x.size() // length(x) // vector size
% C.rows() // size(C,1) // number of rows
% C.cols() // size(C,2) // number of columns
% x(i) // x(i+1) // Matlab is 1-based
% C(i,j) // C(i+1,j+1) //
%
% A.resize(4, 4); // Runtime error if assertions are on.
% B.resize(4, 9); // Runtime error if assertions are on.
% A.resize(3, 3); // Ok; size didn't change.
% B.resize(3, 9); // Ok; only dynamic cols changed.
%
% A << 1, 2, 3, // Initialize A. The elements can also be
% 4, 5, 6, // matrices, which are stacked along cols
% 7, 8, 9; // and then the rows are stacked.
% B << A, A, A; // B is three horizontally stacked A's.
% A.fill(10); // Fill A with all 10's.
%
% // Eigen // Matlab
% MatrixXd::Identity(rows,cols) // eye(rows,cols)
% C.setIdentity(rows,cols) // C = eye(rows,cols)
% MatrixXd::Zero(rows,cols) // zeros(rows,cols)
% C.setZero(rows,cols) // C = zeros(rows,cols)
% MatrixXd::Ones(rows,cols) // ones(rows,cols)
% C.setOnes(rows,cols) // C = ones(rows,cols)
% MatrixXd::Random(rows,cols) // rand(rows,cols)*2-1 // MatrixXd::Random returns uniform random numbers in (-1, 1).
% C.setRandom(rows,cols) // C = rand(rows,cols)*2-1
% VectorXd::LinSpaced(size,low,high) // linspace(low,high,size)'
% v.setLinSpaced(size,low,high) // v = linspace(low,high,size)'
% VectorXi::LinSpaced(((hi-low)/step)+1, // low:step:hi
% low,low+step*(size-1)) //
%
%
% // Matrix slicing and blocks. All expressions listed here are read/write.
% // Templated size versions are faster. Note that Matlab is 1-based (a size N
% // vector is x(1)...x(N)).
% /******************************************************************************/
% /* PLEASE HELP US IMPROVING THIS SECTION */
% /* Eigen 3.4 supports a much improved API for sub-matrices, including, */
% /* slicing and indexing from arrays: */
% /* http://eigen.tuxfamily.org/dox-devel/group__TutorialSlicingIndexing.html */
% /******************************************************************************/
% // Eigen // Matlab
% x.head(n) // x(1:n)
% x.head<n>() // x(1:n)
% x.tail(n) // x(end - n + 1: end)
% x.tail<n>() // x(end - n + 1: end)
% x.segment(i, n) // x(i+1 : i+n)
% x.segment<n>(i) // x(i+1 : i+n)
% P.block(i, j, rows, cols) // P(i+1 : i+rows, j+1 : j+cols)
% P.block<rows, cols>(i, j) // P(i+1 : i+rows, j+1 : j+cols)
% P.row(i) // P(i+1, :) 取第i+1行
% P.col(j) // P(:, j+1) 取第j+1行
% P.leftCols<cols>() // P(:, 1:cols)
% P.leftCols(cols) // P(:, 1:cols)
% P.middleCols<cols>(j) // P(:, j+1:j+cols)
% P.middleCols(j, cols) // P(:, j+1:j+cols)
% P.rightCols<cols>() // P(:, end-cols+1:end)
% P.rightCols(cols) // P(:, end-cols+1:end)
% P.topRows<rows>() // P(1:rows, :)
% P.topRows(rows) // P(1:rows, :)
% P.middleRows<rows>(i) // P(i+1:i+rows, :)
% P.middleRows(i, rows) // P(i+1:i+rows, :)
% P.bottomRows<rows>() // P(end-rows+1:end, :)
% P.bottomRows(rows) // P(end-rows+1:end, :)
% P.topLeftCorner(rows, cols) // P(1:rows, 1:cols)
% P.topRightCorner(rows, cols) // P(1:rows, end-cols+1:end)
% P.bottomLeftCorner(rows, cols) // P(end-rows+1:end, 1:cols)
% P.bottomRightCorner(rows, cols) // P(end-rows+1:end, end-cols+1:end)
% P.topLeftCorner<rows,cols>() // P(1:rows, 1:cols)
% P.topRightCorner<rows,cols>() // P(1:rows, end-cols+1:end)
% P.bottomLeftCorner<rows,cols>() // P(end-rows+1:end, 1:cols)
% P.bottomRightCorner<rows,cols>() // P(end-rows+1:end, end-cols+1:end)
%
% // Of particular note is Eigen's swap function which is highly optimized.
% // Eigen // Matlab
% R.row(i) = P.col(j); // R(i, :) = P(:, j)
% R.col(j1).swap(mat1.col(j2)); // R(:, [j1 j2]) = R(:, [j2, j1])
%
% // Views, transpose, etc;
% /******************************************************************************/
% /* PLEASE HELP US IMPROVING THIS SECTION */
% /* Eigen 3.4 supports a new API for reshaping: */
% /* http://eigen.tuxfamily.org/dox-devel/group__TutorialReshape.html */
% /******************************************************************************/
% // Eigen // Matlab
% R.adjoint() // R' 共轭转置 (对于复数来说,会进行一次共轭操作)
% R.transpose() // R.' or conj(R') 转置 // conj共轭(conjugate)
% R.diagonal() // diag(R) // Read-write
% x.asDiagonal() // diag(x)
% R.transpose().colwise().reverse() // rot90(R) // Read-write
% R.rowwise().reverse() // fliplr(R)
% R.colwise().reverse() // flipud(R)
% R.replicate(i,j) // repmat(P,i,j)
%
%
% // All the same as Matlab, but matlab doesn't have *= style operators.
% // Matrix-vector. Matrix-matrix. Matrix-scalar.
% y = M*x; R = P*Q; R = P*s;
% a = b*M; R = P - Q; R = s*P;
% a *= M; R = P + Q; R = P/s;
% R *= Q; R = s*P;
% R += Q; R *= s;
% R -= Q; R /= s;
%
% // Vectorized operations on each element independently
% // 独立地对每个元素进行矢量化操作
% // Eigen // Matlab
% R = P.cwiseProduct(Q); // R = P .* Q
% R = P.array() * s.array(); // R = P .* s
% R = P.cwiseQuotient(Q); // R = P ./ Q
% R = P.array() / Q.array(); // R = P ./ Q
% R = P.array() + s.array(); // R = P + s
% R = P.array() - s.array(); // R = P - s
% R.array() += s; // R = R + s
% R.array() -= s; // R = R - s
% R.array() < Q.array(); // R < Q
% R.array() <= Q.array(); // R <= Q
% R.cwiseInverse(); // 1 ./ P
% R.array().inverse(); // 1 ./ P
% R.array().sin() // sin(P)
% R.array().cos() // cos(P)
% R.array().pow(s) // P .^ s
% R.array().square() // P .^ 2
% R.array().cube() // P .^ 3
% R.cwiseSqrt() // sqrt(P)
% R.array().sqrt() // sqrt(P)
% R.array().exp() // exp(P)
% R.array().log() // log(P)
% R.cwiseMax(P) // max(R, P)
% R.array().max(P.array()) // max(R, P)
% R.cwiseMin(P) // min(R, P)
% R.array().min(P.array()) // min(R, P)
% R.cwiseAbs() // abs(P)
% R.array().abs() // abs(P)
% R.cwiseAbs2() // abs(P.^2)
% R.array().abs2() // abs(P.^2)
% (R.array() < s).select(P,Q ); // (R < s ? P : Q)
% R = (Q.array()==0).select(P,R) // R(Q==0) = P(Q==0)
% R = P.unaryExpr(ptr_fun(func)) // R = arrayfun(func, P) // with: scalar func(const scalar &x);
%
%
% // Reductions.
% // 弱化;缩减量.
% int r, c;
% // Eigen // Matlab
% R.minCoeff() // min(R(:))
% R.maxCoeff() // max(R(:))
% s = R.minCoeff(&r, &c) // [s, i] = min(R(:)); [r, c] = ind2sub(size(R), i);
% s = R.maxCoeff(&r, &c) // [s, i] = max(R(:)); [r, c] = ind2sub(size(R), i);
% R.sum() // sum(R(:))
% R.colwise().sum() // sum(R)
% R.rowwise().sum() // sum(R, 2) or sum(R')'
% R.prod() // prod(R(:))
% R.colwise().prod() // prod(R)
% R.rowwise().prod() // prod(R, 2) or prod(R')'
% R.trace() // trace(R)
% R.all() // all(R(:))
% R.colwise().all() // all(R)
% R.rowwise().all() // all(R, 2)
% R.any() // any(R(:))
% R.colwise().any() // any(R)
% R.rowwise().any() // any(R, 2)
%
% // Dot products, norms, etc.
% // 点积，范数等.
% // Eigen // Matlab
% x.norm() // norm(x). Note that norm(R) doesn't work in Eigen.
% x.squaredNorm() // dot(x, x) Note the equivalence is not true for complex
% x.dot(y) // dot(x, y)
% x.cross(y) // cross(x, y) Requires #include <Eigen/Geometry>
%
% //// Type conversion
% //// 类型强制转换
% // Eigen // Matlab
% A.cast<double>(); // double(A)
% A.cast<float>(); // single(A)
% A.cast<int>(); // int32(A)
% A.real(); // real(A)
% A.imag(); // imag(A)
% // if the original type equals destination type, no work is done
%
% // Note that for most operations Eigen requires all operands to have the same type:
% MatrixXf F = MatrixXf::Zero(3,3);
% A += F; // illegal in Eigen. In Matlab A = A+F is allowed
% A += F.cast<double>(); // F converted to double and then added (generally, conversion happens on-the-fly)
%
% // Eigen can map existing memory into Eigen matrices.
% float array[3];
% Vector3f::Map(array).fill(10); // create a temporary Map over array and sets entries to 10
% int data[4] = {1, 2, 3, 4};
% Matrix2i mat2x2(data); // copies data into mat2x2
% Matrix2i::Map(data) = 2*mat2x2; // overwrite elements of data with 2*mat2x2
% MatrixXi::Map(data, 2, 2) += mat2x2; // adds mat2x2 to elements of data (alternative syntax if size is not know at compile time)
%
% // Solve Ax = b. Result stored in x. Matlab: x = A \ b or inv(A)*b.
% x = A.ldlt().solve(b)); // A sym. p.s.d. #include <Eigen/Cholesky>
% x = A.llt() .solve(b)); // A sym. p.d. #include <Eigen/Cholesky>
% x = A.lu() .solve(b)); // Stable and fast. #include <Eigen/LU> ★☆
% x = A.qr() .solve(b)); // No pivoting. #include <Eigen/QR>
% x = A.svd() .solve(b)); // Stable, slowest. #include <Eigen/SVD>
% // .ldlt() -> .matrixL() and .matrixD()
% // .llt() -> .matrixL()
% // .lu() -> .matrixL() and .matrixU()
% // .qr() -> .matrixQ() and .matrixR()
% // .svd() -> .matrixU(), .singularValues(), and .matrixV()
%
% // Eigenvalue problems
% // 特征值问题
% // Eigen // Matlab
% A.eigenvalues(); // eig(A);
% EigenSolver<Matrix3d> eig(A); // [vec val] = eig(A)
% eig.eigenvalues(); // diag(val)
% eig.eigenvectors(); // vec
% // For self-adjoint matrices use SelfAdjointEigenSolver<>
%% ----上面已完结，下面是Eigen库关于矩阵和向量的一些预定义 typedef ----
% ★★☆☆ 常规的矩阵typedef ☆☆★★
% 我们后面给出了一些常用的矩阵typedef.其实可以总结如下：
%
% MatrixNt对应的是Matrix<type,N,N>.比如MatrixXi对应的是Matrix<int,Dynamic,Dynamic>.
% VectorNt对应的是Matrix<type,N,1>.比如Vector2f对应的是Matrix<float,2,1>.
% RowVectorNt对应的是Matrix<type,1,N>.比如RowVector3d对应的是Matrix<double,1,3>.
% 其中:
%
% N可以是2,3,4或者X(表示Dynamic).
% t可以是i(int),f(float),d(double),cf(complex),cd(complex).
% 只定义了这些类型的typedef并不表示只支持这些数据类型的运算。
% 比如所有的整形类型的运算都支持（长的，短的，有符号的，无符号的）。
% typedef Matrix< std::complex<double> , 2 , 2 > Matrix2cd
% typedef Matrix< std::complex<float> , 2 , 2 > Matrix2cf
% typedef Matrix< double , 2 , 2 > Matrix2d
% typedef Matrix< float , 2 , 2 > Matrix2f
% typedef Matrix< int , 2 , 2 > Matrix2i
% typedef Matrix< std::complex<double> , 3 , 3 > Matrix3cd
% typedef Matrix< std::complex<float> , 3 , 3 > Matrix3cf
% typedef Matrix< double , 3 , 3 > Matrix3d
% typedef Matrix< float , 3 , 3 > Matrix3f
% typedef Matrix< int , 3 , 3 > Matrix3i
% typedef Matrix< std::complex<double> , 4 , 4 > Matrix4cd
% typedef Matrix< std::complex<float> , 4 , 4 > Matrix4cf
% typedef Matrix< double , 4 , 4 > Matrix4d
% typedef Matrix< float , 4 , 4 > Matrix4f
% typedef Matrix< int , 4 , 4 > Matrix4i
% typedef Matrix< std::complex<double> , Dynamic , Dynamic > MatrixXcd
% typedef Matrix< std::complex<float> , Dynamic , Dynamic > MatrixXcf
% typedef Matrix< double , Dynamic , Dynamic > MatrixXd
% typedef Matrix< float , Dynamic , Dynamic > MatrixXf
% typedef Matrix< int , Dynamic , Dynamic > MatrixXi
% typedef Matrix< std::complex<double> , 1, 2 > RowVector2cd
% typedef Matrix< std::complex<float> , 1, 2 > RowVector2cf
% typedef Matrix< double , 1, 2 > RowVector2d
% typedef Matrix< float , 1, 2 > RowVector2f
% typedef Matrix< int , 1, 2 > RowVector2i
% typedef Matrix< std::complex<double> , 1, 3 > RowVector3cd
% typedef Matrix< std::complex<float> , 1, 3 > RowVector3cf
% typedef Matrix< double , 1, 3 > RowVector3d
% typedef Matrix< float , 1, 3 > RowVector3f
% typedef Matrix< int , 1, 3 > RowVector3i
% typedef Matrix< std::complex<double> , 1, 4 > RowVector4cd
% typedef Matrix< std::complex<float> , 1, 4 > RowVector4cf
% typedef Matrix< double , 1, 4 > RowVector4d
% typedef Matrix< float , 1, 4 > RowVector4f
% typedef Matrix< int , 1, 4 > RowVector4i
% typedef Matrix< std::complex<double> , 1, Dynamic > RowVectorXcd
% typedef Matrix< std::complex<float> , 1, Dynamic > RowVectorXcf
% typedef Matrix< double , 1, Dynamic > RowVectorXd
% typedef Matrix< float , 1, Dynamic > RowVectorXf
% typedef Matrix< int , 1, Dynamic > RowVectorXi
% typedef Matrix< std::complex<double> , 2 , 1> Vector2cd
% typedef Matrix< std::complex<float> , 2 , 1> Vector2cf
% typedef Matrix< double , 2 , 1> Vector2d
% typedef Matrix< float , 2 , 1> Vector2f
% typedef Matrix< int , 2 , 1> Vector2i
% typedef Matrix< std::complex<double> , 3 , 1> Vector3cd
% typedef Matrix< std::complex<float> , 3 , 1> Vector3cf
% typedef Matrix< double , 3 , 1> Vector3d
% typedef Matrix< float , 3 , 1> Vector3f
% typedef Matrix< int , 3 , 1> Vector3i
% typedef Matrix< std::complex<double> , 4 , 1> Vector4cd
% typedef Matrix< std::complex<float> , 4 , 1> Vector4cf
% typedef Matrix< double , 4 , 1> Vector4d
% typedef Matrix< float , 4 , 1> Vector4f
% typedef Matrix< int , 4 , 1> Vector4i
% typedef Matrix< std::complex<double> , Dynamic , 1> VectorXcd
% typedef Matrix< std::complex<float> , Dynamic , 1> VectorXcf
% typedef Matrix< double , Dynamic , 1> VectorXd
% typedef Matrix< float , Dynamic , 1> VectorXf
% typedef Matrix< int , Dynamic , 1> VectorXi
